Etale Homotopy
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100 M. Artin · B. Mazur
Etale Homotopy
Springer-Verlag Berlin Heidelberg New York Tokyo
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
100 M. Artin · B. Mazur
Etale Homotopy
Springer-Verlag Berlin Heidelberg New York Tokyo
Authors Michael Artin Massachusetts Institute of Technology Rm 2-239, Cambridge, MA 02139, USA Barry Mazur Harvard University One Oxford St., Cambridge, MA 02138, USA
1st Edition 1969 2nd Printing 1986
ISBN 3-540-04619-4 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-04619-4 Springer-Verlag New York Heidelberg Berlin Tokyo
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© by Springer-Verlag Berlin Heidelberg 1969 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Contents
................. ...................
1.
A glossary of the categories in which we shall work, and fibre resolutions.
2.
Pro-objects in the homotopy category••••••••••••••••••••••• 20
3.
Completions ••
4.
Cohomological criteria for
5.
Completions and fib rations.
6.
Homotopy groups of completions ••••••••••••••••••••••••••••• 70
7.
Stable results.
6
....... .... ................... ..... ..... ..... 25
g -isomorphism••••••••••••••••• 35 ......... .. .... ....... ....... .. 60
8.
........................................... 75 Hypercove rings. ........................................... 93 The Verdier functor •••• ............................... ••••• 111
10.
The fundamental group••••••••••.••••••••••••••••••••••••••• 117
11.
A profiniteness theorem•••••••••••••••••••••••••••••••••••• 124
12.
Comparison theorems •••••••••••••••••••••••••••••••••••••••• 129
Appendix: 1.
Lim1 ts •.................................................... 147
2.
Pro-objects and pro-representable functors.
3.
Morphisms
4.
Exactness
............... .154 of pro-objects ••••••••••••••••• . ...... ..........•159 properties of the pro-category. ................. .163
References .................................•.....•.........•... 167
- 1 -
These notes are an expansion of the results announced in [2].
The material was presented in a seminar at Harvard
University during the academic year 165- 166.* Our aim is to study the analogues of homotopy invariants which can be obtained from varieties by using the etale topology of Grothendieck.
Using the constructions of Lubkin [21]
or Verdier [3], we associate to any locally noetherian prescheme
X a pro-object in the homotopy category of simplicial
sets (cf. 2), which we call the etale homotopy prescheme
X.
of the
For a normal variety over the field of complex
numbers, we show that
Xe t is a certain profinite c
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